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In the survey paper Poset Topology: Tools and Applications by Michelle Wachs, there is the following theorem on p46:

Theorem 3.2.4 (Björner and Wachs [40]). Suppose $P$ is a poset for which $\hat{P}$ admits an EL-labeling. Then $P$ has the homotopy type of a wedge of spheres, where the number of $i$-spheres is the number of decreasing maximal $(i+2)$-chains of $\hat{P}$.

The notation $\hat{P}$ means $ P \cup \{\hat{0}, \hat{1}\}$. The homotopy type of a poset $P$ is the homotopy type of the geometric realization of its order complex $\Delta(P)$.

Now let $P$ be a poset of order $n+1$ and length $0$ (i.e. just isolated points), then $\hat{P}$ admits an obvious EL-labeling with $n$ decreasing maximal $2$-chains. So by Theorem 3.2.4, $P$ should have the homotopy type of a wedge of $n$ number of $0$-sphere, i.e just one point, but it is not correct because it is a set of $n+1$ points.

I would need some clarifications about this inconsistency, in particular:.

Question: Is the statement of Theorem 3.2.4 correct if we assume that there is no isolated point?

Remark: If a poset admits an EL-labeling then it is Cohen-Macaulay (see here), and a poset is Cohen-Macaulay iff its Stanley–Reisner ring is a Cohen-Macaulay ring.

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    $\begingroup$ Each zero sphere is a collection of 2 points, (one basepoint and one other point) and when you wedge $n$ of them together, you identify the basepoints, yielding $n+1$ points in total. $\endgroup$
    – Phil Tosteson
    Apr 20, 2016 at 12:11
  • $\begingroup$ @Phil: Ok, I had a false belief about a 0-sphere. I just post mathoverflow.net/a/236771/34538 $\endgroup$
    – Sebastien Palcoux
    Apr 20, 2016 at 12:41

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